Sh. Ismailov, A. Qo’chqorov, B. Abdurahmoi

pq

A
< 3 tengsizlikni isbotlang.
gar n e N bo’lsa, 2 <

A
ⁿ

n

< n!
gar n > 0 bo’lsa, quyidagi tengsizlikni isbotlang

.

3n

21. 
    Agar n > 0 bo’lsa, quyidagi tengsizlikni isbotlang. (3n)! >(n)
    
22. 
    Agar ^ = a₁ + a₂ +... + a_n; a_t > 0, i = 1,2,.,n bo’lsa,
    

s S s

(1 + a₁ )(1 + a₂)-. (1 + a_n)< 1 ^ —+ — +... + — tengsizlikni isbotlang.

1! 2! n

!
a

b)

a

1

• 
  —
  

21. 
    a) — „
    

a + a +1 3

. ₂ 1 1

s) a + —7 > a +—;

a

21. 
    a) 9a² _ 30|a| + 25 > 0; s) a² _ 4a + 5 > 2 a _ 2|;
    
22. 
    a) a⁴ + b⁴ > a³b + ab³;
    

s) a⁶ + b⁶ > a⁴b² + a²b⁴:

a1

< —

a² _ 4a + 9 2

14 a⁴ +16 „ d) —= > 2a. a² + 4 b) b² + 25 > 10 |b|; d) b² _ 2b +10 > 6 |b _ 1|.

b) a⁴ + b⁴ + ab(a² + b²) > 0; d) a⁶ + b⁶ > a⁵b + ab⁵

.

21. 
    Agar a > 0 va b > 0 bo’lsa, u holda quyidagilarni isbotlang:
    

a) a³ + b³ > a²b + ab²; b) (a + b)³ < 4(a³ + b³);

s) a⁵ + b⁵ > a⁴b + ab⁴; d) a⁵ + b⁵ > a³b² + a²b³.

21. 
    Agar a > 0 va b > 0 bo’lsa, u holda quyidagilarni isbotlang:
    

a) a³ _ b³ > a²b _ ab²; b) a³ _ b³ > 3ab(a _ b);

s) a³ _ b³ > ab² _ a²b; d) a⁵ _ b⁵ > a⁴b _ ab⁴.

21. 
    a va b sonlarning ixtiyoriy qiymatlarida tengsizlik o’rinli
    

bo’lishini isbotlang:

1. 
   a⁴ _ 2a³b + 2a²b² _ 2ab³ + b⁴ > 0;
   
2. 
   a⁴ _ 4a³b + 8a²b² _ 16ab³ + 16b⁴ > 0.
   

21. 
    Ixtiyoriy a, b, c, va d sonlar uchun tengsizlik
    

1. 
   (a² _ b² )(c² _ d²) < (ac _ bd)² ;
   
2. 
   (a² + b² )(c² + d²) > (ac + bd)².
   

o’rinli bo’lishini isbotlang, jumladan tenglik bajariladi shu holda va faqat shu holdaki, qachonki ad = bc

21. 
    .ab > 0 shartni qanoatlantiruvchi ixtiyoriy a va b sonlar uchun (a² - b² )² > (a - b)⁴ tengsizlik o’rinli bo’lishini isbotlang.
    
22. 
    Agar a < b bo’lsa, a < ~~~ < b bo’lishini isbotlang.
    

a + b + c

21. 
    Agar a < b < c bo’lsa, a < 3 < b bo’lishini isbotlang.
    
22. 
    Agar a > 0, b > 0, c < 0, d < 0 ekanligi ma’lum.